Journal of Experimental Psychology:
Animal Behavior Processes
2011, Vol. 37, No. 2, 246 –253
© 2011 American Psychological Association
0097-7403/11/$12.00 DOI: 10.1037/a0021215
BRIEF REPORT
Orientation in Trapezoid-Shaped Enclosures:
Implications for Theoretical Accounts of Geometry Learning
Bradley R. Sturz
Taylor Gurley and Kent D. Bodily
Armstrong Atlantic State University
Georgia Southern University
Human participants learned to select 1 of 4 distinctively marked corners in a rectangular virtual enclosure.
After training, control and test trials were interspersed with training trials. On control and test trials, all markers
were equivalent in color, but only during test trials was the shape of the enclosure manipulated. Specifically,
for each test trial, a single long wall or short wall of the enclosure increased twice as long as or decreased half
as long as that present in the training enclosure. These manipulations produced 8 unique trapezoid-shaped
enclosures. Participants were allowed to select 1 corner during control and test trials. Performance during
control trials revealed that participants selected the correct and rotationally equivalent locations. Performance
during test trials revealed that participants selected locations in trapezoid-shaped enclosures that were
consistent with those predicted by global geometry (i.e., principal axis of space) but were inconsistent with
those predicted by local geometry (i.e., proportion of rewarded training features present at a location). Results
have implications for theoretical accounts of geometry learning.
Keywords: virtual environment, global geometry, local geometry, orientation, trapezoid
virtual environments (Kelly & Bischof, 2005, 2008), and dynamic
three-dimensional virtual environments (Alexander, Wilson, &
Wilson, 2009; Sturz & Diemer, 2010; Sturz & Kelly, 2009; Wilson
& Alexander, 2008). Regardless of the species under investigation
or the preparation used, the ubiquitous result is that organisms
trained to select a particular corner in a rectangular enclosure
containing distinct cues at each corner allocated equivalent responses to the correct and rotationally equivalent corners when
distinct cues were removed or rendered identical during test trials.
In all cases, responses to these geometrically correct corners were
significantly greater than those to the geometrically incorrect corners (for a review, see Cheng & Newcombe, 2005).
Such results have been taken as evidence for the encoding of
global environmental shape (Cheng, 1986; for a review, see Cheng
& Newcombe, 2005), and theoretical explanations of this geometry learning suggest that global geometry is extracted from enclosures incidentally by a dedicated geometric module (Cheng, 1986;
Gallistel, 1990; for a review, see Cheng & Newcombe, 2005).
According to global-based accounts of geometry learning, participants extract the principal axis of space in the enclosure, and this
principal axis coupled with a sense (i.e., left or right) or distance
component guides the participant to both the correct and rotationally equivalent locations during cue-absent trials because these
locations become indistinguishable via this strategy in the absence
of the distinct cue present at the correct location (Cheng, 1986;
Cheng & Gallistel, 2005; Cheng & Newcombe, 2005; Gallistel,
1990; for a review, see Cheng, 2005; see also Doeller & Burgess,
2008; Doeller, King, & Burgess, 2008). Such a global geometry
account (i.e., principal axis-based) seems to explain many observed phenomena in enclosure search tasks such as continued
responding to geometrically correct locations when enclosure size
is manipulated between training and testing (Sovrano, Bisazza, &
Over the past three decades, orientation by means of the shape
of an enclosure has been demonstrated in a variety of species
including rats (Cheng, 1986), chickens (Vallortigara, Zanforlin, &
Pasti, 1990), pigeons (Kelly, Spetch, & Heth, 1998), fish (Sovrano,
Bisazza, & Vallortigara, 2003), monkeys (Gouteux, Thinus-Blanc,
& Vauclair, 2001), and ants (Wystrach & Beugnon, 2009). Most
germane for present purposes, orientation via enclosure shape has
been obtained with both human children (Hermer & Spelke, 1994;
Learmonth, Newcombe, & Huttenlocher, 2001) and adults (Hartley, Trinkler, & Burgess, 2004; Hermer-Vazquez, Spelke, &
Katsnelson, 1999) in such diverse preparations as real environments (Hermer-Vazquez et al., 1999), static three-dimensional
This article was published Online First February 14, 2011.
Bradley R. Sturz, Department of Psychology, Armstrong Atlantic State
University; Taylor Gurley and Kent D. Bodily, Department of Psychology,
Georgia Southern University.
This research was conducted following the relevant ethical guidelines
for human research and partially supported by a National Science Foundation Science and Technology Expansion Program Grant (NSF-STEP
DUE-0856593) to the College of Science and Technology at Armstrong
Atlantic State University and an Armstrong Atlantic State University
Summer Research Grant to Bradley R. Sturz. We thank Raymond J. Spiteri
for the calculations of the principal axes of space for the trapezoid-shaped
enclosures along with Ken Cheng and an anonymous reviewer for comments on a previous version of the article. We are indebted to Ken Cheng
for bringing to our attention model-selection techniques and its application
to geometry learning. Please note that first and third authors contributed
equally to this project.
Correspondence concerning this article should be addressed to Bradley
R. Sturz, Armstrong Atlantic State University, Department of Psychology,
229 Science Center, 11935 Abercorn Street, Savannah, GA 31419. E-mail:
[email protected]
246
ORIENTATION IN TRAPEZOID-SHAPED ENCLOSURES
Vallortigara, 2007; Sturz & Kelly, 2009) and immunity of global
geometry to cue competition from landmark learning (Cheng,
1986; for a review, see Cheng & Newcombe, 2005).
More recent theoretical proposals have suggested that the correct location is decomposed into its component parts (i.e., features
such as long wall left, short wall right, 90° angle), which are
rewarded along with the cue present at that location, and according
to these local geometry (i.e., feature-based) accounts of geometry
learning, responses in enclosures are allocated to the location with
the largest number of training features present divided by the sum
total of all the other training features present in that trial (Dawson,
Kelly, Spetch, & Dupuis, 2010; Miller, 2009; Miller & Shettleworth, 2007; Ponticorvo & Miglino, 2010). These features then
guide the participant to both the correct and rotationally equivalent
locations during cue-absent trials because these locations also
become indistinguishable via this strategy in the absence of the
distinctive cue at the correct location due to that fact that the
number of rewarded features present at each of these two corners
is equivalent. Such a local geometry account (i.e., feature-based
account) seems to explain many observed phenomena in enclosure
search tasks such as differential influences (i.e., cue competition or
lack thereof) of features or geometry and differential influences of
features or geometry in differently sized enclosures (Miller, 2009;
Miller & Shettleworth, 2007).
The purpose of the present experiment was to provide an explicit
test of predictions derived from global-based and local-based accounts
of geometry learning (i.e., Cheng, 1986; Dawson et al., 2010; Gallistel, 1990; Miller & Shettleworth, 2007; Ponticorvo & Miglino, 2010).
The present experiment attempted to discriminate between these two
general classes of theoretical accounts of geometry learning but not to
discriminate between any specific iteration from either of the respective classes of models. To that end, we trained human participants to
select one of four distinctively marked corners (i.e., red, yellow, blue,
or green cues) in a rectangular virtual enclosure. After training, we
presented control and test trials in which all cues were rendered
identical (i.e., white). Control trials occurred in the rectangular enclosure used during training, whereas test trials occurred in manipulations of the shape of the rectangular enclosure used during training.
Specifically, for each test trial, a single long wall or short wall of the
rectangular training enclosure increased twice as long or decreased
half as long to create eight unique trapezoid-shaped enclosures (see
Figure 1).
The unique shapes of the trapezoid-shaped enclosures allowed us
to explicitly place predictions derived from global-based and localbased accounts of geometry learning in conflict for theoretical diagnostic purposes. Given the nature of the trapezoid-shaped enclosures
used in the present experiment, the principal axis always specified the
same two locations (i.e., top left and bottom right) as those of the
training and control trials, but these locations differed with respect to
the number of local features present compared with that of training
and control trials. For example, a location in the test enclosure may
have contained any number of features present during training, ranging from zero to three via a combination of a long wall to the left, a
short wall to the right, and a 90° angle.
According to global-based accounts of geometry learning, responses should be allocated equally to the two locations dictated by
the principal axis of space. As a result, within our trapezoid-shaped
enclosures, the principal axis should guide search equivalently to the
top left and bottom right locations. In contrast, according to local-
247
based accounts of geometry learning, responses should be allocated to
those locations that contain the largest proportions of rewarded training features. Within our trapezoid-shaped enclosures, location with
respect to the principal axis was in conflict with the proportion of
rewarded training features present at a location.
Method
Participants
Thirty-two Georgia Southern University undergraduate students
(16 men and 16 women) served as participants. Participants received extra class credit.
Apparatus
An interactive, dynamic three-dimensional virtual environment
was constructed and rendered using Valve Hammer Editor and run
on the Half-Life Team Fortress Classic platform. A personal
computer, 21-in. flat-screen liquid crystal display monitor, gamepad joystick, and speakers served as the interface with the virtual
environment. The monitor (1,152 ⫻ 864 pixels) provided a firstperson perspective of the virtual environment (see top panel,
Figure 1). The joystick on the gamepad navigated within the
environment. Speakers emitted auditory feedback. Experimental
events were controlled and recorded using the Half-Life Dedicated
Server on an identical personal computer.
Stimuli
Dimensions are length ⫻ width ⫻ height and measured in virtual
units (vu). Ten virtual enclosures were created (see Figure 1): rectangle (552 ⫻ 276 ⫻ 260 vu), control rectangle (552 ⫻ 276 ⫻ 260
vu), Trapezoids 1– 8 (552 ⫻ 138 ⫻ 260 vu; 552 ⫻ 276 ⫻ 260 vu;
552 ⫻ 552 ⫻ 260 vu; 1104 ⫻ 276 ⫻ 260 vu). Each enclosure
contained four orbs (48 ⫻ 48 ⫻ 48 vu) arranged one in each corner.
Orb colors were red, blue, yellow, green, or white depending on trial
type (see below). Walls were textured with beige concrete and the
floors with gray tiles. Ceilings were pure black.
Procedure
Participants were instructed to navigate to the orb that transported them to the next virtual room and moved via the joystick on
the gamepad: 1 (forward), 2 (backward), 4 (left), and 3
(right). Participants selected an orb by walking into it. Selection of
the rewarded corner resulted in auditory feedback (bell sound) and
a 7-s intertrial interval in which the monitor went black and
participants progressed to the next trial. Selection of a nonrewarded corner resulted in different auditory feedback (buzz sound)
and required participants to continue searching.
Training. Training consisted of eight trials. Participants were
randomly assigned to one of the four corners, which was then
designated as the rewarded corner. Gender and number of participants trained at each corner were balanced. Participants started
each trial in the center of the rectangle (marked with a gray circle
in Figure 1). Participants entered the rectangle at random orientations from 0° to 270° in increments of 90°. Each of the four corners
was marked with a distinct color: blue, yellow, red, or green.
STURZ, GURLEY, AND BODILY
248
Training Trial
3[.76]
.53
3[.76]
.09
.13 .25
.25
1[.25]
.06
.50
.50
.19
0 [.0]
.25
.56
2[.51]
.06
.27
.36
Above Chance
Equal to Chance
Below Chance
.44
.11
.13 .41
2[.51]
3 [.76]
.03
1 [.25]
.05
.20
0 [.0]
3[.76]
.45
Test Trial
2 [.51]
2[.51]
.06
.27
.09
.11
0 [.0]
.11
.22
.39
Control
.34
.48
.09
1 [.25]
.30
1 [.25]
.16
.14
.42
0 [.0]
Figure 1. Sample images from the first-person perspective (top left and top right) of virtual environment search
spaces. Schematics, predicted proportion of responses for global-based accounts (0.38 to top left and bottom right
corners) and local-based (brackets outside trapezoid-shaped enclosures) accounts of geometry learning, and
mean proportion of responses to each corner for each test enclosure (bold inside trapezoid-shaped
enclosures). Note that the number to the left of the brackets for local-based predictions indicates the number
of rewarded training features (i.e., long wall left, short wall right, 90° angle) present at that location. Also
note that the predicted proportions of responses were averaged for top left and bottom right locations as
shown in the bottom panel of Figure 2. Finally, note that these predicted proportions of responses were used
to calculate the residuals for the Akaike information criterion (see text of Results section for details). For
illustrative purposes, the gray circles mark the position where participants entered the virtual enclosures for all training and
testing trials. Dotted lines represent the principal axis of space for each enclosure.
Testing. Testing consisted of 54 trials composed of 18 threetrial blocks. Each trial block was composed of two training trials
and one test trial. Location of the test trial was randomized within
block. For each test trial, one of nine enclosures was presented:
control, trapezoids 1– 8. Each enclosure was presented once without replacement until all nine had been presented. Each enclosure
was presented two times (total of 18 test trials). Participants made
one response during test trials, which resulted in no auditory
feedback followed by the 7-s intertrial interval and progression to
the next trial. Participants entered all enclosures during testing in
the center of the enclosures (marked with gray circles in Figure 1)
at random orientations from 0° to 270° in increments of 90°. All
orbs were white during test trials.
Results
Training
Figure 2 (top panel) shows the mean proportion of participants’
first responses to the rewarded corner plotted by two-trial blocks
for the eight trials of training. As shown, participants rapidly
learned to respond to the rewarded corner. A two-way mixed
ORIENTATION IN TRAPEZOID-SHAPED ENCLOSURES
249
Mean Proportion of Correct First Responses
1.0
0.8
0.6
0.4
Chance
0.2
0.0
Block 1
Block 2
Block 3
Block 4
Two-Trial Blocks
1.0
Mean Proportion of Responses
Obtained
Local-Based (Feature-Based) Predicted
0.8
0.6
Global-Based
(Principal Axis)
Predicted
0.4
Chance
0.2
0.0
Top Left
Bottom Right
Trapezoid-Shaped Enclosure Location
Figure 2. (Top panel) Mean proportion of participants’ first responses to the rewarded corner plotted by
two-trial blocks for the eight trials of training. (Bottom panel) Mean proportion of obtained (hashed bars) and
local geometry predicted (unfilled bars) responses to the top left and bottom right trapezoid-shaped enclosure
locations. Dotted line represents the proportion of responses predicted by global geometry. Dashed lines
represent chance performance. Error bars represent standard errors of the mean.
analysis of variance on mean proportion of first responses to the
rewarded corner with gender (male, female) and block (1– 4) as
factors revealed only a main effect of block, F(3, 90) ⫽ 59.74, p ⬍
.001. Neither the effect of gender nor the interaction was significant, Fs ⬍ 1, ps ⬎ .57. In addition, all blocks were significantly
greater than chance performance (i.e., 0.25), ts(31) ⬎ 2.7, ps ⬍ .05
(one-sample t tests). Given the rapid acquisition of the task, we
wanted to ensure that initial performance (i.e., Trial 1) was at
chance (i.e., 0.25). The proportion of participants that selected the
rewarded corner as their first response during Trial 1 (0.28) was
not significantly different from what would be expected by chance,
␹2(1, N ⫽ 32) ⫽ 0.17, p ⬎ .68.
Testing
For data analytic purposes, results from all participants were
adjusted to be as if they had been trained at top left or bottom right
locations. This was accomplished by transposing the mean proportion of responses to each location with the mirror equivalent for
participants trained in top right or bottom left locations. As a
result, figures reflect responses as if all participants had been
trained at the top left location. Figure 1 (bottom) shows the mean
proportion of responses to each corner for each enclosure collapsed across both presentations of each enclosure (numbers inside
enclosures). In the control rectangle, in the absence of distinct
cues, the mean proportion of responses to the correct and rotationally equivalent corners did not differ from each other, t(31) ⫽
⫺0.71, p ⬎ .48 (paired-samples t test), and the mean proportion of
responses allocated to these geometrically correct corners (M ⫽
0.76, SEM ⫽ 0.06) was greater than would be expected by chance
(i.e., 0.5), t(31) ⫽ 4.5, p ⬍ .001 (one-sample t test). For trapezoidshaped enclosures, we compared the mean proportion of responses
to each corner for each enclosure type to chance performance (i.e.,
250
STURZ, GURLEY, AND BODILY
0.25). Correcting for multiple comparisons, Figure 1 (bottom) also
delineates the mean proportions that were significantly above
(underlined), below (italicized), ts(31) ⬎ 2.3 ps ⬍ .01 (one-sample
t tests), and equal to (normal font) chance (i.e., 0.25), ts(31) ⬍ 1.7,
ps ⬎ .01 (one-sample t tests).
We also compared the obtained mean proportion of responses in
trapezoid-shaped enclosures with the mean proportion of responses in the control enclosure. The obtained mean proportion of
responses to the top left and bottom right locations on the control
trials (M ⫽ 0.76, SEM ⫽ 0.06) were not statistically different from
top left and bottom right locations on the trapezoid-shaped enclosures (M ⫽ 0.77, SEM ⫽ 0.03), t(31) ⫽ ⫺0.07, p ⬎ .95 (paired t
test). As with performance in the control enclosure, the obtained
mean proportion of responses to top left and bottom right locations
in the trapezoid-shaped enclosures was greater than chance (0.5),
one sample t test, t(31) ⫽ 9.18, p ⬍ .001.
Predictions.
Global-based accounts predict equivalent responding to the top left and bottom right locations because a
strategy of following the principal axis and then searching at the
left-hand side designates these two locations as indistinguishable
(Cheng, 1986; Gallistel, 1990; see also Cheng & Gallistel, 2005;
for a review, see Cheng, 2005). However, local-based accounts
predict that the allocation of responses to these two locations
should be a function of the number of training features present at
each corner relative to the total number of training features at those
two locations (see Dawson et al., 2010; Miller, 2009; Miller &
Shettleworth, 2007; Ponticorvo & Miglino, 2010). Despite differences in these predictions regarding allocation of responses to a
location, both models suggest that their respective predictions
should hold for both average results across shape manipulations
and for each specific trapezoid-shaped enclosure (see Figure 1). As
a result, we used both approaches to assist in discriminating
between these theoretical accounts of geometry learning.
Performance in orientation tasks is rarely allocated exclusively
to correct and rotationally equivalent corners during control trials,
possibly due to factors such as random error, errors in encoding, or
errors in retrieval. The present results are no exception; therefore,
we adopted a more conservative approach to calculating predictions for these models of geometry learning. Specifically, we based
predictions for both models relative to obtained baseline performance in the control enclosure. For global-based accounts, we
calculated the prediction of the proportion of responses to each the
top left and bottom right locations as 0.38 (baseline performance at
top left and bottom right [i.e., 0.76] divided by 2). In contrast, for
local geometry, we separately calculated the prediction of the
proportion of responses to the top left and bottom right locations
by counting the number of training features present at each location (i.e., long wall left, short wall right, and 90° angle; the number
of features present at top left and bottom right locations is shown
to the left of the brackets in Figure 1) and dividing this number by
the total number of training features at both locations ([number of
training features present at a location/number of total training
features present at top left and bottom right locations] ⫻ baseline
performance at top left and bottom right locations [i.e., 0.76]). The
resulting value was the predicted proportion of responses to that
location (number inside brackets in Figure 1). Such a method was
employed separately for each of these locations for each trapezoidshaped enclosure. It should be noted that these three independent
features (i.e., long wall left, short wall right, and 90° angle) were
assumed to be equally weighted binary features; however, assumptions based on training reinforcement values of these features (i.e.,
50% for long wall and short wall and 25% for 90° angle) do not
alter the predictions.
Figure 2 (bottom panel) shows the predictions for a globalbased (principal axis-based) account (dotted line) and for a
local-based (feature-based) account (unfilled bars). As shown,
global-based and local-based models predict differences in responding to both the top left location (one-sample t test comparing
top left local-based predictions to 0.38), t(7) ⫽ 2.39, p ⬍ .05, and
the bottom right location (one-sample t test comparing bottom
right local-based predictions to 0.38), t(7) ⫽ ⫺2.39, p ⬍ .05.
Global-based accounts predict equivalent responding to both top
left and bottom right locations, whereas local-based accounts predict preferential responding to the top left location relative to the
bottom right location (paired-samples t test comparing local-based
predictions for top left and bottom right), t(7) ⫽ 2.39, p ⬍ .05. In
addition, global-based accounts predict responding to both top left
and bottom right locations each at above chance levels (i.e., 0.25),
whereas local-based accounts predict that the top left location
should be above chance (one-sample t test comparing top left
local-based predictions to chance), t(7) ⫽ 4.03, p ⬍ .01, whereas
the bottom right location should be at chance (one-sample t test
comparing bottom right local-based predictions to chance), t(7) ⫽
⫺0.76, p ⬎ .47.
Comparisons of predicted to obtained. To determine how
well global geometry (principal axis-based) and local geometry
(feature-based) predictions fit obtained data, we compared the
obtained proportions of responses with predictions from these
models. The obtained mean proportions of responses to the top left
and bottom right locations were not statistically different from that
predicted by global-based accounts (0.38), t(7) ⫽ 0.97, p ⬎ .36,
and t(7) ⫽ ⫺0.77, p ⬎ .46, respectively (one-sample t tests). The
obtained mean proportions of responses to the top left and bottom
right locations were also not statistically different from that predicted by local-based accounts, t(14) ⫽ 1.67, p ⬎ .1, and t(14) ⫽
⫺1.8, p ⬎ .09, respectively (independent samples t tests). However, as predicted by global-based but not local-based accounts,
the obtained data showed no difference between the mean proportion of responses to top left and bottom right locations, t(7) ⫽ 0.89,
p ⬎ .4 (paired t test). Moreover, as predicted by global geometry
but not local geometry, the obtained mean proportions of responses
to the top left and bottom right locations were each greater than
chance (0.25), t(7) ⫽ 4.14, p ⬍ .01, and t(7) ⫽ 2.55, p ⬍ .05
(one-sample t tests).
We acknowledge that the average predictions for each class of
model as analyzed above may not apply to each specific trapezoidshaped enclosure presented in Figure 1. As a result, we also used
the Akaike information criterion (AIC) to compare predictions
derived from global-based and local-based accounts of geometry
learning to obtained data (see Burnham & Anderson, 2002). The
AIC is an alternative to hypothesis testing based on modelselection techniques that assists in determining how well a particular model fits obtained data. Based on mean error2, the AIC
allows models to be ranked according to how well they fit obtained
data. Specifically, AIC ⫽ nlog(error) ⫹ 2(r ⫹ 2), where n ⫽ the
number of data points, r ⫽ the number of free parameters, and
ORIENTATION IN TRAPEZOID-SHAPED ENCLOSURES
error ⫽ mean error2 (n – r – 1)/n. In short, the model with the
lowest (i.e., most negative) AIC value is considered the “best” fit
(for detailed application to spatial learning, see Narendra, Cheng,
Sulikowski, & Wehner, 2008).
Figure 1 summarizes the predictions derived from global-based
and local-based accounts of geometry learning. We calculated the
mean error for each model by subtracting the obtained proportion
of responses to the top left and bottom right (i.e., geometrically
correct) locations from the predicted proportion of responses to
these locations for each enclosure and averaged these residuals.
For global-based accounts, the mean error2 for the top left location
was 0.0016 and for the bottom right location was 0.0009. For
local-based accounts, the mean error2 for the top left location was
0.0225 and for the bottom right location was 0.0256. Using the
formula above, we calculated AIC values for each model. For
global-based accounts, AIC values for top left and bottom right
locations were ⫺18.83 and ⫺20.83, respectively. For local-based
account, AIC values for top left and bottom right locations were
⫺9.65 and ⫺9.19, respectively. As a result, the AIC values suggest
that global-based accounts of geometry learning fit the obtained
data better than local-based accounts of geometry learning. This
analysis is consistent with the results from hypothesis testing
reported above and provides converging evidence to suggest that
the obtained results are more consistent with global-based compared with local-based accounts of geometry learning.
Discussion
Results in the present dynamic three-dimensional virtual environment search task appear consistent with extant human and
nonhuman animal research conducted in real environment enclosures that document the rotational error phenomenon (for a review,
see Cheng & Newcombe, 2005). Specifically, during the control
trials when the distinct training cue was absent, participants allocated equivalent responses to the correct and rotationally equivalent corners, and responses to these locations were statistically
greater than responses to the other two locations. In addition,
present results extend previous research by demonstrating that in
novel enclosures in which the principal axis and proportion of
training features present at a location conflict, participants responded as if following the principal axis. Such a result appears
consistent with global-based (Cheng, 1986; Gallistel, 1990; for a
review, see Cheng & Newcombe, 2005) but not local-based accounts of geometry learning (e.g., Dawson et al., 2010; Miller,
2009; Miller & Shettleworth, 2007; Ponticorvo & Miglino, 2010).
We acknowledge that our approach may not be the most sophisticated approach to distinguishing between the two classes of
models, especially given the fact that there are a host of additional
assumptions coupled with additional parameters one could include
in any one model to improve the specificity of its predictions (e.g.,
weighting values, relative vs. absolute metrics, distances from the
principal axis, etc.). However, our minimalistic approach includes
what is both sufficient and necessary for each class of model to
maintain its theoretical distinctiveness while still being able to
explain the fundamental phenomenon (i.e., the rotational error) it
was intended to explain. As a result, data analyzed in the current
fashion not only distinguish between the two classes of models
because the categorical assumptions made by each class of models
251
result in clearly divergent predictions regarding search location
during the test trials, but also the obtained results are diagnostic of
such predictions. Minimally, the current results suggest that localbased accounts of geometry learning using binary coding of the
presence or absence of features are insufficient to account for
responses in the present task.
In remains unclear whether the principal axis could be incorporated into a local geometry model (i.e., Dawson et al., 2010; Miller
& Shettleworth, 2007; Ponticorvo & Miglino, 2010) by treating the
principal axis as an additional local feature (albeit an unperceived
one), but such a hypothesis is consistent with a recent empirical
and theoretical focus on the relative weighting of spatial cues
(Cheng, Shettleworth, Huttenlocher, & Rieser, 2007; Nardini,
Jones, Bedford, & Braddick, 2008; Newcombe & Ratliff, 2007;
Ratliff & Newcombe, 2008; see also Nardini, Thomas, Knowland,
Braddick, & Atkinson, 2009; Newcombe, Ratliff, Shallcross, &
Twyman, 2010). This approach may potentially explain the difficulty in local geometry models’ ability to explain control by
geometry when enclosure size is manipulated between training and
testing (Miller, 2009; see also Kelly & Spetch, 2001; Sovrano et
al., 2007; Sturz & Kelly, 2009) and results of experiments in which
features (e.g., colors of walls) move randomly during training (see
Graham, Good, McGregor, & Pearce, 2006). Such an approach
may also potentially explain the difficulty of global geometry
models’ ability to explain the results of experiments in which
competition/facilitation between global geometry and features is
obtained (Gray, Bloomfield, Ferrey, Spetch, & Sturdy, 2005; Kelly,
2010; for reviews, see Cheng, 2008; Twyman & Newcombe, 2010;
see also Doeller & Burgess, 2008; Doeller et al., 2008).
It also remains unclear how an alternative model of geometry
learning, view-based matching, could account for present results.
A view-based matching account of geometry learning suggests that
the enclosure is stored as a representation from the goal location in
memory and involves reducing the discrepancy between an organism’s current retinal image and this stored representation. Responding is suggested to be determined by the best match of
current perception with this stored representation. Although ants,
honeybees, and human infants appear to use such a view-based
matching strategy (for reviews, see Cheng, 2000, 2008; see also
Cheung, Stürzl, Zeil, & Cheng, 2008; Nardini et al., 2009; Stürzl,
Cheung, Cheng, & Zeil, 2008; Wystrach & Beugnon, 2009), there
is recent evidence against the use of a view-based matching
strategy in human adults (e.g., Nardini et al., 2009; Sturz &
Diemer, 2010). In addition, participant responses in the trapezoidshaped enclosures of the current experiment were oftentimes allocated at above-chance levels to locations that were not a best visual
match of the contours of the training enclosure (refer to Figure 1).
In conclusion, when participants were trained in a rectangular
enclosure and tested with trapezoid-shaped enclosures in which the
principal axis and proportion of rewarded training features present
at a location were in conflict, they responded in these novel
enclosures as if following the principal axis. As a result, obtained
data are consistent with global-based (i.e., principal axis-based)
but not local-based (feature-based) accounts of geometry learning.
Although it is unclear whether the principal axis could be treated
as a feature and then weighted relative to all other features present
in the enclosure (or potentially averaged with available features
when in conflict; see Cheng et al., 2007; Nardini et al., 2009;
STURZ, GURLEY, AND BODILY
252
Newcombe & Ratliff, 2007), it seems clear that local-based accounts of geometry learning using binary coding of the presence or
absence of features are insufficient to explain the current results.
Future research could explore these and other theoretical assumptions coupled with additional parameters to inform theoretical
accounts of geometry learning and illuminate the mechanisms
underlying orientation with respect to the environment.
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Received May 5, 2010
Revision received August 4, 2010
Accepted August 9, 2010 䡲
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